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Mirrors > Home > ILE Home > Th. List > tfi | Unicode version |
Description: The Principle of
Transfinite Induction. Theorem 7.17 of [TakeutiZaring]
p. 39. This principle states that if is a class of ordinal
numbers with the property that every ordinal number included in
also belongs to , then every ordinal number is in .
(Contributed by NM, 18-Feb-2004.) |
Ref | Expression |
---|---|
tfi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2311 | . . . . . . 7 | |
2 | imdi 239 | . . . . . . . 8 | |
3 | 2 | albii 1359 | . . . . . . 7 |
4 | 1, 3 | bitri 173 | . . . . . 6 |
5 | dfss2 2934 | . . . . . . . . . 10 | |
6 | 5 | imbi2i 215 | . . . . . . . . 9 |
7 | 19.21v 1753 | . . . . . . . . 9 | |
8 | 6, 7 | bitr4i 176 | . . . . . . . 8 |
9 | 8 | imbi1i 227 | . . . . . . 7 |
10 | 9 | albii 1359 | . . . . . 6 |
11 | 4, 10 | bitri 173 | . . . . 5 |
12 | simpl 102 | . . . . . . . . . . 11 | |
13 | tron 4119 | . . . . . . . . . . . . . 14 | |
14 | dftr2 3856 | . . . . . . . . . . . . . 14 | |
15 | 13, 14 | mpbi 133 | . . . . . . . . . . . . 13 |
16 | 15 | spi 1429 | . . . . . . . . . . . 12 |
17 | 16 | spi 1429 | . . . . . . . . . . 11 |
18 | 12, 17 | jca 290 | . . . . . . . . . 10 |
19 | 18 | imim1i 54 | . . . . . . . . 9 |
20 | impexp 250 | . . . . . . . . 9 | |
21 | impexp 250 | . . . . . . . . . 10 | |
22 | bi2.04 237 | . . . . . . . . . 10 | |
23 | 21, 22 | bitri 173 | . . . . . . . . 9 |
24 | 19, 20, 23 | 3imtr3i 189 | . . . . . . . 8 |
25 | 24 | alimi 1344 | . . . . . . 7 |
26 | 25 | imim1i 54 | . . . . . 6 |
27 | 26 | alimi 1344 | . . . . 5 |
28 | 11, 27 | sylbi 114 | . . . 4 |
29 | 28 | adantl 262 | . . 3 |
30 | sbim 1827 | . . . . . . . . . 10 | |
31 | clelsb3 2142 | . . . . . . . . . . 11 | |
32 | clelsb3 2142 | . . . . . . . . . . 11 | |
33 | 31, 32 | imbi12i 228 | . . . . . . . . . 10 |
34 | 30, 33 | bitri 173 | . . . . . . . . 9 |
35 | 34 | ralbii 2330 | . . . . . . . 8 |
36 | df-ral 2311 | . . . . . . . 8 | |
37 | 35, 36 | bitri 173 | . . . . . . 7 |
38 | 37 | imbi1i 227 | . . . . . 6 |
39 | 38 | albii 1359 | . . . . 5 |
40 | ax-setind 4262 | . . . . 5 | |
41 | 39, 40 | sylbir 125 | . . . 4 |
42 | dfss2 2934 | . . . 4 | |
43 | 41, 42 | sylibr 137 | . . 3 |
44 | 29, 43 | syl 14 | . 2 |
45 | eqss 2960 | . . 3 | |
46 | 45 | biimpri 124 | . 2 |
47 | 44, 46 | syldan 266 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wal 1241 wceq 1243 wcel 1393 wsb 1645 wral 2306 wss 2917 wtr 3854 con0 4100 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-in 2924 df-ss 2931 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 |
This theorem is referenced by: tfis 4306 |
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